Solve 2(x-1)^2+2=0

$$2 ( x - 1 ) ^ { 2 } + 2 = 0$$

Solve for x (complex solution)

$$x=1-i$$

$$x=1+i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2\left(x^{2}-2x+1\right)+2=0$$

Use the distributive property to multiply 2 by x^{2}-2x+1.

$$2x^{2}-4x+2+2=0$$

Add 2 and 2 to get 4.

$$2x^{2}-4x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 4}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\times 4}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\times 4}}{2\times 2}$$

Multiply -8 times 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-32}}{2\times 2}$$

Add 16 to -32.

$$x=\frac{-\left(-4\right)±\sqrt{-16}}{2\times 2}$$

Take the square root of -16.

$$x=\frac{-\left(-4\right)±4i}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±4i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±4i}{4}$$

Now solve the equation x=\frac{4±4i}{4} when ± is plus. Add 4 to 4i.

$$x=\frac{4+4i}{4}$$

Divide 4+4i by 4.

$$x=1+i$$

Now solve the equation x=\frac{4±4i}{4} when ± is minus. Subtract 4i from 4.

$$x=\frac{4-4i}{4}$$

Divide 4-4i by 4.

$$x=1-i$$

The equation is now solved.

$$x=1+i x=1-i$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2\left(x^{2}-2x+1\right)+2=0$$

Use the distributive property to multiply 2 by x^{2}-2x+1.

$$2x^{2}-4x+2+2=0$$

Add 2 and 2 to get 4.

$$2x^{2}-4x+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-4x=-4$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=-\frac{4}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=-\frac{4}{2}$$

Divide -4 by 2.

$$x^{2}-2x=-\frac{4}{2}$$

Divide -4 by 2.

$$x^{2}-2x=-2$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-2+1$$

Add -2 to 1.

$$x^{2}-2x+1=-1$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x-1=i x-1=-i$$

Add 1 to both sides of the equation.

$$x=1+i x=1-i$$